Spectral link of the generalized Townsend-Perry constants in turbulent boundary layers
SSpectral link of the generalized Townsend-Perry constants in turbulent boundarylayers
Björn Birnir , Luiza Angheluta , John Kaminsky , Xi Chen CNLS and Department of Mathematics, University of California in Santa Barbara, USA The Njord Centre, Department of Physics, University of Oslo, P. O. Box 1048, 0316 Oslo, Norway Institute of Fluid Mechanics, Beihang University, Beijing, China
We propose a minimal spectral theory for boundary layer turbulence that captures very well theprofile of the mean square velocity fluctuations in the stream-wise direction, and gives a quantitativeprediction of the Townsend-Perry constants. The phenomenological model is based on connectingthe statistics in the streamwise direction with the energy spectrum of the streamvise velocity fluc-tuations. The original spectral theory was proposed in Ref. [8] to explain the friction factor andvon Kármán law in Ref. [7]. We generalized it by including fluctuations in the wall-shear stress andthe streamwise velocity. The predicted profiles for the mean velocity and mean square fluctuationsare compared with velocity data from wind tunnel experiments.
I. INTRODUCTION
Turbulence is a ubiquitous phenomenon encounteredin very diverse natural systems, from the large-scale at-mosphere [27] and oceans [23] all the way down to quan-tum fluids [26], as well as in engineered systems, such aspipelines, heat exchangers, wind turbines, etc. It relatesto the complex fluid dynamics that orchestrates the in-teractions of flow eddies spanning many length-scales andgenerating non-Gaussian statistics of velocity increments.The statistical properties of these turbulent fluctuationsare fundamentally changed when the flow is confined bythe presence of solid walls or boundaries [9, 22]. In con-trast to bulk turbulence, which is statistically homoge-neous and isotropic, the wall-bounded turbulence is char-acterised by statistically anisotropic properties. Namely,there is a net mean-flow in the streamwise direction alongthe wall and the different flow structures form dependingon their distance to the wall. We typically differentiatebetween four flow regions as moving away from the wall[17]: i) the viscous region closest to the wall and whereviscous flows dominate, ii) the buffer layer , marking thetransition from the viscous layer into the inertial layer,iii) the inertial layer where the log-law of the wall ap-plies, and iv) the wake , the energetic region beyond theinertial layer. A more refined division is given in [6].A classical signature of wall-bounded turbulence is the"log-law of the wall" of the mean velocity profile (MVP)due to Prandtl and von Kármán, and reads as (cid:104) ˜ u (cid:105) = 1 κ log(˜ y ) + B, (1)where κ is the universal von Kármán constant that isindependent of the microscopic flow characteristics andrelates to generic features such as space dimensionality.The distance to the wall y and the mean fluid velocity u along the wall, are typically expressed in the "wall units"determined by the wall shear stress τ . This is because τ is an important theoretical concept that is also experi-mentally measurable. The friction velocity u τ = (cid:112) (cid:104) τ (cid:105) /ρ which is set by the wall shear stress τ and the kinematic < w > y < u > Viscous Buffer Inertial
FIG. 1. Theoretical predictions from the spectral theory forthe MVP (cid:104) u (cid:105) and mean square velocity fluctuations (cid:104) w (cid:105) (di-mensionless variables in wall units). viscosity ν , and enters in the unit rescalings as ˜ u = u/u τ and ˜ y = yu τ /ν . The constant fluid density is ρ and the B is a dimensionless constant that is fitted to experimentaldata, e.g. [20].A log-law of the wall was also derived from the "at-tached eddy hypothesis" by Townsend [24]. Townsendshowed that the velocity fluctuations, ˜ w = w/u τ , ˜ u = (cid:104) ˜ u (cid:105) + ˜ w , also follow the log-law of the wall in its secondmoment, namely (cid:104) ˜ w (cid:105) = − A log(˜ y ) + B , (2)where the coefficients A and B , also called theTownsend-Perry constants, were first measured by Perryand Chong [18, 19].More recently, the log-law was generalised to any mo-ment of the streamwise velocity fluctuations, ˜ w , assumingGaussian velocity fluctuations [15], (cid:104) ˜ w p (cid:105) /p = − A p log(˜ y ) + B p . (3)While the generalised log-law is supported by wall-turbulence experiments, the dependance of A p and B p on p turns out to be sub-Gaussian, which is confirmed both a r X i v : . [ phy s i c s . f l u - dyn ] J un experimentally and numerically, [15]. The sub-Gaussianbehavior was explained in Ref. [4] using the stochasticclosure theory of turbulence [2, 3] and the analysis wasimproved in Ref. [12], using measurements from the FlowPhysics Facility (FPF) at the University of New Hamp-shire. Both of these studies used the results from homo-geneous turbulence [11] and made an assumption aboutthe form of the fluctuating shear stress in the inertiallayer, based on physical principles.In Ref. [7], a spectral theory for the log-law of thewall of the MVP was proposed in which it is possible toderive the log-law in the inertial layer and the laminarprofile in the viscous layer. The novel contribution is theprecise form of the transition in the buffer layer usingthe the Kolmogorov-Obukhov energy spectrum of turbu-lent fluctuations. The form of the MVP in the wake isalso obtained. This was done by summing the energyof the wall-attached eddies, as hypothesised originally byTownsend in [24].In this paper, we propose a generalisation of the spec-tral theory that includes fluctuations in the streamwisevelocity due to an essentially fluctuating wall shear stress.Fig. 5 shows the spectral theory predictions of the profilesof the mean velocity and mean square velocity fluctua-tions across the viscous, buffer and inertial layers. Therest of the paper is structured as follow. We summarisethe analysis in Ref. [7] and its extension in Section II,and generalise it to include the fluctuations in SectionIII. This produces the log law of the wall in Eq. (2) forthe velocity fluctuations and its higher moments in Eq.(3). Then in Section IV, we derive the functional formof the mean-square fluctuations in the viscous layer andthe inertial layer. In Section V, we use the attached eddyhypothesis and the stochastic closure theory [2, 3] to de-rive the form of the Townsend-Perry and the generalizedTownsend-Perry constants. This allows us to derive thestreamwise fluctuations in the wall shear stress, and re-move the assumption made in Refs. [4] and [12], andmentioned above. Using theory-informed by data analy-sis, we can construct the Townsend-Perry constants andthe generalised Townsend-Perry constants. In SectionVI, we extend the formulas for the mean square fluc-tuations to the buffer layer and the energetic wake. InSection VII, we compare the predicted MVP and mean-square velocity profile from this spectral theory to ex-perimental data. In Section VIII, we conclude with adiscussion on the proposed spectral theory and the rolethat Townsend’s attached eddies play in it. II. THE SPECTRAL THEORY
The typical velocity of an inertial eddy of size s can beobtained by integrating out the kinetic energy containedin all eddies of sizes up to s as in Ref. [7] v s = (cid:90) ∞ /s E ( k ) dk, (4) where kinetic energy spectrum follows the Kolmogorov-Obukhov scaling with cutoffs in the injection scale andviscous scales, E ( k ) = c d ( ηk ) ( κ (cid:15) (cid:15) ) / k − / c e ( Rk ) , with ( κ (cid:15) (cid:15) ) / k − / being the Kolmogorov-Obukhov spec-trum and c d ( ηk ) and c e ( Rk ) the phenomenological di-mensionless corrections functions in the dissipative (setby the Kolmogorov scale η ) and energetic range (set bythe system size R ), respectively. κ (cid:15) is a dimensionlessparameter, (cid:15) is the turbulent energy dissipation rate, η = ν / (cid:15) − / is the viscous length scale and R is thelargest length scale in the flow. The dissipative correc-tion function is typically an exponential cutoff function c d ( ηk ) = exp( − β d ηk ) , and the energetic-range (wake)correction function is c e ( Rk ) = (1 + ( β e / ( Rk )) ) − / ,which is the form that was proposed by von Kármán. β d and β e are non-negative fitting parameters that can beadjusted to data. By the change of variables ξ = sk , werecast Eq. (4) as v s = ( κ (cid:15) (cid:15)s ) / I (cid:16) ηs , sR (cid:17) , (5)where the spectral function I is given by the formula [7] I (cid:16) ηs , sR (cid:17) =23 (cid:90) ∞ e − ξβ d η/s ξ − / (cid:32) (cid:18) β e sRξ (cid:19) (cid:33) − / dξ. (6)The integral sums the energies of all eddies of a smallerradius than s , and computes their contribution to theenergy of the eddy of radius s . This is the energy (orspectral) formulation of the attached eddy hypothesis ofTownsend [24]. The I -function correctly captures thebuffer layer, as the transition from the viscous to theinertial layer, and the asymptotic of the MVP in the en-ergetic wake. The asymptotic values are such that in theinertial layer I = 1 and in the viscous layer I = 0 . The I -function combines the Kolmogorov-Obukhov theory withthe observed spectrum in the viscous layer, the inertiallayer and the wake and is thus able to capture the tran-sition from one layer to the next. In Ref [7], it was usedto give the details of the MVP. In this paper, we will useit to capture the profile of mean-square fluctuations.In the buffer layer a different scaling of the attachededdies comes into play, this is the k − x scaling of the spec-trum that has been debated in literature, but clearlyshows up in recent simulations and experiments in themiddle of the buffer layer, see Figure 9 (a) in Ref. [14]and Figure 12 (b) in Ref. [21]. In the spectral theory,corresponding I -function for this scaling regime is I b (cid:16) ηs , sR (cid:17) =23 s − (cid:90) ∞ e − ξβ d ηs ξ − (cid:32) (cid:18) β e sRξ (cid:19) (cid:33) − dξ, (7)where the subscript b stands for "buffer". The mean ve-locity is primarily influenced by the I -function, whereasthe variation (fluctuation squared) is greatly influencedby the I b -function in the buffer layer. I is associated withthe Kolmogorov-Obukhov energy cascade k − / x , in theinertial layer, whereas I b is associated with the k − x scal-ing in the buffer layer. (Here the x denotes the stream-wise direction.) We will take I b to be zero outside thebuffer layer.The splitting of the near-wall region based on differentscaling of the spectrum was proposed by Perry and Chong[18] who used it build an interpolation model for MVPand the variation, this model was improved in Ref. [25]. III. THE GENERALISED LOG-LAW
In this section, we will give a simple derivation of thelog-law for the mean-square velocity profile that holdsin the limit of large Reynolds number. In the followingsection we derive the general form of the variation thatis not equally transparent.We will generalize the derivation of the MVP in Ref.[7], by adding a fluctuation to the mean velocity. We letthe velocity along the wall be v = u + v − u = u + w, (8)where u is the mean velocity obtained by averaging v over time, and w is the fluctuation. The same derivationsas in Ref. [7] give the following equations for a dominanteddy of radius s = y , if we include the velocity fluctua-tions. In Ref. [7] the shear stress at the distance y fromthe wall is given by the formula ¯ τ t = κ τ ρyv y u (cid:48) where u (cid:48) denotes the y derivative of the velocity u along thewall, and the overline indicates a not-fluctuating quan-tity. When velocity fluctuations are included the shearstress becomes: τ t = κ τ ρyv y ( u (cid:48) + w (cid:48) ) , (9)where ρ is the density v y is the (rotational) velocity of aneddy a distance y from the wall and κ τ is the dimension-less proportionality factor. The energy dissipation rateis related to the wall shear stress as ¯ (cid:15) = τ t u (cid:48) /ρ [7] , andincluding the fluctuations, this becomes (cid:15) = τ t ( u (cid:48) + w (cid:48) ) /ρ. (10)The eddy velocity for an eddy with radius s = y at thedistance y from the wall is the same as in Ref. [7], andas discussed above, v y = ( κ (cid:15) (cid:15)y ) / √ I, (11)where I is the integral from Eq. (6) and κ (cid:15) is a dimen-sionless proportionality factor. In the inertial layer I = 1 and κ (cid:15) = 4 / according to Kolmogorov’s / law.Eliminating (cid:15) and v y from the three equations above,we obtain τ t = ( κ (cid:15) κ τ ) / ρy ( u (cid:48) + w (cid:48) ) I / . (12) The viscous shear stress is ρν ( u (cid:48) + w (cid:48) ) so the total shearstress, including the contribution from the fluctuation is[24] τ t + ρν ( u (cid:48) + w (cid:48) ) = τ (1 − y/R ) . (13)Our assumption is that the wall shear stress τ is also aquantity that fluctuates about its mean value.We change the rescaled variables in the wall units writ-ten here in terms of the friction factor f : ˜ y = yRe √ f /R , ˜ u = u/ ( U √ f ) and ˜ w = w/ ( U √ f ) and let f = (cid:104) τ (cid:105) /ρU .Then, the equation above becomes ˜ κ ˜ y (˜ u (cid:48) + ˜ w (cid:48) ) I / +(˜ u (cid:48) + ˜ w (cid:48) ) = τ (cid:104) τ (cid:105) (cid:18) − ˜ yRe √ f (cid:19) . (14)If we let ˜ y → , ˜ w → and integrate, we get the law ofthe viscous layer ˜ u = ˜ y, (15)the laminar profile being ˜ u = (cid:18) ˜ y − ˜ y Re √ f (cid:19) . (16)In the large Reynolds number limit, solving just for themean velocity, we obtain the Prandtl-von Kármán law ˜ u = 1˜ κ log(˜ y ) + D. (17)This is the correct leading term but the full formulas inthe next section are more complicated. We now motivatethe log-law for the variation. If we solve for both themean velocity and the fluctuation in the large Reynoldsnumber limit, we get that ˜ u + ˜ w = √ τ (cid:104) τ (cid:105) / ˜ κ log(˜ y ) + C. (18)This is consistent with the Eq. (17) in the sense thatif √ τ = (cid:104) τ (cid:105) / , then ˜ w = 0 and we recover Eq. (17).Thus squaring Eq. (18) gives that ˜ u +2˜ u ˜ w + ˜ w = τ (cid:104) τ (cid:105) ˜ κ (log(ˆ y )) +2 √ τ ˜ κ (cid:112) (cid:104) τ (cid:105) C log(˜ y )+ C . (19)Taking the average, using that (cid:104) ˜ w (cid:105) = 0 and Eq. (17), weget that (cid:104) ˜ w (cid:105) = 2 C (cid:104)√ τ (cid:105) − D (cid:112) (cid:104) τ (cid:105) ˜ κ (cid:112) (cid:104) τ (cid:105) log(˜ y ) + C − D . (20)By comparing this with the generalised log-law in Eq.(2), for the fluctuations squared, we obtain (cid:104) ˜ w (cid:105) = − A log(˜ y ) + B, (21)where A = − C (cid:104)√ τ (cid:105)− D √ (cid:104) τ (cid:105) ˜ κ √ (cid:104) τ (cid:105) and B = C − D arethe Townsend-Perry constants. The full formulas in nextsection show that Eq. (21) is the leading term and A = − C ( (cid:104)√ τ (cid:105)− √ (cid:104) τ (cid:105) ˜ κ √ (cid:104) τ (cid:105) ) , with C = D .To simplify the notation, we will now drop the tilde’sfrom all the variable with the dimensionless units implic-itly assumed, unless otherwise stated. IV. THE FUNCTIONAL FORM OF THETOWNSEND-PERRY LAW
We will now use Eq. (14) to find the general form ofthe average of the fluctuations squared as a function ofthe distance to the wall. We consider the Eq. (14) κ y ( u (cid:48) + w (cid:48) ) I / + ( u (cid:48) + w (cid:48) ) = τ (cid:104) τ (cid:105) (1 − yRe √ f ) , (22)and first set I = 0 in the viscous layer. Then u = y − y Re √ f (23)by averaging and integration in y . Integrating Eq. (22)and subtracting u gives, w = τ − (cid:104) τ (cid:105)(cid:104) τ (cid:105) (cid:18) y − y Re √ f (cid:19) (24)and (cid:104) w (cid:105) = (cid:104) τ (cid:105) − (cid:104) τ (cid:105) (cid:104) τ (cid:105) (cid:18) y − y Re √ f (cid:19) . (25)In the inertial layer I = 1 and ignoring the small O (1 /y ) term, we get that u + w = 12 κ y + 2 √ τ κ (cid:112) (cid:104) τ (cid:105) (cid:114) − y Re √ f − √ τ κ (cid:112) (cid:104) τ (cid:105) tanh − (cid:18)(cid:114) − y Re √ f (cid:19) + K, (26)where K is a constant. Then setting w = 0 , we get that u = 12 κ y + 2 κ (cid:114) − y Re √ f − κ tanh − (cid:18)(cid:114) − y Re √ f (cid:19) + K (cid:48) , (27)where K (cid:48) is another constant, because τ becomes (cid:104) τ (cid:105) .Subtracting, u from u + w we get w = 2 ( √ τ − (cid:112) (cid:104) τ (cid:105) ) κ (cid:112) (cid:104) τ (cid:105) (cid:114) − y Re √ f − √ τ − (cid:112) (cid:104) τ (cid:105) ) κ (cid:112) (cid:104) τ (cid:105) tanh − (cid:18)(cid:114) − y Re √ f (cid:19) + C, (28) where C = K − K (cid:48) . Squaring w and taking the averagegives (cid:104) w (cid:105) = 4 C ( (cid:104)√ τ (cid:105) − (cid:112) (cid:104) τ (cid:105) ) κ (cid:112) (cid:104) τ (cid:105) (cid:114) − y Re √ f − C ( (cid:104)√ τ (cid:105) − (cid:112) (cid:104) τ (cid:105) ) κ (cid:112) (cid:104) τ (cid:105) tanh − (cid:18)(cid:114) − y Re √ f (cid:19) + 4 (cid:34) (cid:104) τ (cid:105) − (cid:112) (cid:104) τ (cid:105)(cid:104)√ τ (cid:105) ) κ (cid:104) τ (cid:105) (cid:18) − y Re √ f − (cid:114) − y Re √ f tanh − ( (cid:114) − y Re √ f ) (cid:19) + (cid:20) tanh − ( (cid:114) − y Re √ f ) (cid:21) (cid:35) + C . (29)From tanh − ( x ) = log( x − x ) , we see that the secondterm in the last formula is of leading order and we getthat (cid:104) w (cid:105) ∼ C ( (cid:104)√ τ (cid:105) − (cid:112) (cid:104) τ (cid:105) ) κ (cid:112) (cid:104) τ (cid:105) log (cid:18) yRe √ f (cid:19) + h.o.t. (30)This agrees with the formula (21) above. For higher ordermoments (cid:104) w p (cid:105) /p the similar term, linear in tanh − andmultiplied by C , is of leading order, (cid:104) w p (cid:105) /p ∼ C (cid:104) ( √ τ − (cid:112) (cid:104) τ (cid:105) ) p (cid:105) /p κ (cid:112) (cid:104) τ (cid:105) log (cid:18) yRe √ f (cid:19) + h.o.t. (31)These formulas establish the log dependance of the sec-ond moment of the fluctuations, with the Townsend-Perry constants, and the log dependence of the highermoments of the fluctuations, with the GeneralizedTownsend-Perry constants, and justify formulas Eq. (2)and Eq. (3). Together, Eq. (2) and Eq. (3) can be calledthe generalised log-law of the wall. V. DERIVATION OF THE GENERALIZEDTOWNSEND-PERRY CONSTANTS
We consider the dependence of the fluctuation w on thedistance x along the wall, to understand the Townsend-Perry constants. So far we have only considered w ( y ) asa function of the distance y from the wall, but w ( x, y ) obviously depends on both variables x and y . If we con-sider the eddy depicted in Fig. 2, then we see that thedifference in momentum in the x direction, across theeddy, is given by ρ ( w ( x + s ) − w ( x − s )) ∼ ρsw x , (32)for y fixed, where w x = ddx w .This means that the total turbulent stress, across avertical surface at x , denoted by a dotted line on Fig. 2for an eddy of radius s ∼ y , is τ = τ t + τ x , (33) FIG. 2. The eddy of radius s and the variation in the fluctu-ations across it in the x (streamwise) direction. where τ x = 2 κ τ ρyw x v y , analogous to formula Eq. (9)above. Then we get, using Eq. (11) and (cid:15) = ( τ t + τ x )( u (cid:48) + w x ) ρ, (34)that τ t + τ x = κ ρI / y ( u (cid:48) + w x ) , (35)where prime denotes the derivative with respect to y , and ( τ t + τ x ) / = κρ / I / y ( u (cid:48) + w x )= (cid:104) τ (cid:105) / + κρ / I / y | w x | , (36)since both parts must be positive. The derivation is com-pletely analogous to the derivation in Sec. III, but herewith w varying in the x direction and w y = 0 . This givesthat for y fixed, τ / − (cid:104) τ (cid:105) / = ( τ t + τ x ) / − (cid:104) τ (cid:105) / = κρ / I / y | w x | . (37)Considering the leading order log( y/ Re √ f ) term in Eq.(30) gives the Townsend-Perry constant A = 2 Cρ / y (cid:104)| w x |(cid:105) (cid:112) (cid:104) τ (cid:105) , (38)and the generalized Townsend-Perry constants A p = 2 Cρ / y (cid:104)| w x | p (cid:105) /p (cid:112) (cid:104) τ (cid:105) , (39)by use of Eq. (31). This justifies the form of the stresstensor assumed in Ref. [4] and used in Ref. [12]. Finally,we get the expressions A = K (cid:104)| w ( x + y ) − w ( x − y ) |(cid:105) (40)and A p = K (cid:104)| w ( x + y ) − w ( x − y ) | p (cid:105) /p , (41) where K is a constant and this produces the relation-ship between the Townsend-Perry and the generalizedTownsend-Perry constants and the structure function ofturbulence, see Ref. [2, 3, 11], used in Ref. [4, 12], A = KC | y ∗ | ζ , (42) A = KC / | y ∗ | ζ / , (43)and A p = KC /pp | y ∗ | ζ p /p , (44)where − y ≤ y ∗ ≤ y . Considering the ratio, washes outthe constant K , A p A = C /pp C / | y ∗ | ζ p /p − ζ / , (45)where the C p s are the Kolmogorov-Obukhov coefficientsof the structure functions from Ref. [2, 3, 11]. The lastratio was used in Ref. [12] to get agreement betweenexperimental data and theory. VI. THE SPECTRAL THEORY OFMEAN-SQUARE FLUCTUATIONS
In the above sections we have not used the spectralinformation in the integral I , in Eq. (6). We have justused the attached eddy hypothesis and set I = 0 in theviscous layer and I = 1 in the inertial layer. But followingRef. [7], we can now use the spectral information throughthe integral I to find the beginning of the buffer layer andthe form of both the MVP u and the fluctuation w in thebuffer layer and in the wake. This allows one to obtainthe full functional form of both u and w as functions ofthe distance y from the wall and compare it with theexperimental data in the next section. By use of theenergy Eq. (10) and the relation η = ν / (cid:15) − / we canfind an expression for η/y , the viscosity parameter thatincreases as we approach the wall y → . If we set thefluctuation equal to zero, η/y = (˜ u (cid:48) (1 − ˜ y/Re (cid:112) f ) − (˜ u (cid:48) ) ) − / ˜ y − (46)and find a formula for ˜ y using this equation along withthe equation κ ˜ y ( u (cid:48) ) I / + u (cid:48) = τ (cid:104) τ (cid:105) (cid:18) − yRe √ f (cid:19) . (47)The resulting formula is given in Ref. [7], ˜ y = (cid:18) ( η/y ) / + κ / I / ( η/y, κ / ( η/y ) / I / ( η/y, (cid:19) . (48)It gives the minimum value of ˜ y for which I ( η/y, > and the small eddies begin to contribute to the turbulent -2 y u Re= 6 10 Re= 10 Re= 1.45 10 Re= 2 10 Theory (a) -2 y u Re= 6 10 Re= 10 Re= 1.45 10 Re= 2 10 Theory (b)
FIG. 3. The average of the MVP as a function of log ( y ) , where y is the distance from the wall. Comparison of experimentaldata with theory (black line). (a) Theoretical curve is givenby an I -integral that interpolates between the k − / x to the k − x with a = 0 . in the buffer region. (b) Theoreticalcurve has a uniform I -integral with the k − / x scaling presentin buffer and inertial regions. shear stress τ t > . In fact for each value of the param-eter β d there is a minimum value of ˜ y denoted ˜ y v belowwhich I = 0 . Only after this minimum does ˜ y increasewith η/y . This gives the end of the viscous layer and thebeginning of the buffer layer and a value of the MVP, u v at ˜ y v . It also gives the value of the fluctuation w at ˜ y v and we can integrate the differential equations for u and w , with respect to y , to get the form of both functions inthe buffer layer, inertial layer and the wake. Along withthe formulas in the viscous layer this gives the full func-tional form. The differential equations use the spectralinformation through the full functional form of I and thetwo parameters β d and β e must be fitted to experimentaldata.Approximations to the MVP and mean square fluctua-tions, based on the formulas in Sec. IV are given in Fig. 3and 4, respectively. To compare with experimental data one must solve the differential equations u (cid:48) = − κ I / y + 1 κI / y (cid:115) − yRe √ f + 14 κ I / y (49)with the initial condition u = 4 . at the beginning ofthe buffer layer y = 4 . . For the fluctuation we firsthave to solve the differential equation, ignoring term oforder O (1 /y ) and higher, w (cid:48) = √ τ − (cid:112) (cid:104) τ (cid:105) κI / y (cid:112) (cid:104) τ (cid:105) (cid:114) − yRe √ f , (50)with the initial condition w = τ −(cid:104) τ (cid:105)(cid:104) τ (cid:105) (cid:16) . − . Re √ f (cid:17) ,from Eq. (24), at the beginning of the buffer layer. Here I ( y ) is the integral in Eq. (6).In practice it is easier to vary the initial conditions thanto change β d and β e , thus we will let the initial condition y o , of w , from Equation (24), vary slightly dependingon the Reynolds number in the simulations below. Theother initial condition w o is given by the formula w o = τ −(cid:104) τ (cid:105)(cid:104) τ (cid:105) (cid:16) y o − y o Re √ f (cid:17) . VII. COMPARISON WITH EXPERIMENTALDATA
The data we use to compare with the theory comesfrom the wind tunnel experiments at the University ofMelbourne using the nano-scale thermal anemometryprobe (NSTAP) to conduct velocity measurements in thehigh Re number boundary layer up to Re τ = 20000 . TheNSTAT has a sensing length almost one order of magni-tude smaller than conventional hot-wire, hence allows fora fully resolved NSTAT measurement of velocity fluctu-ations, [21], [1]. The size of the University of Melbournewind tunnel and the accuracy of the NSTAT permit themeasurement over a very large range of scales. We usethe averaged velocity time-series at Reynolds numbers Re τ = 6000 , , , and the averaged vari-ance at the same Reynolds numbers. Fig. 3 shows themean velocity profiles as a function of normalized dis-tance from the wall, whereas Fig. 4 shows the averagedfluctuation squared (variation) as a function of the nor-malized distance to the wall. Both are semi-log plots.First, let us consider the curve describing the MVP inFig. 3 (panel b). It starts with the Eq. (23) for the vis-cous profile because the I -function is zero. But then wereach the value y v where the first attached eddies appear( y = 4 . ) and then the viscous profile changes, insteadof reaching its maximum u = Re √ f / at y = Re √ f ,the attached eddies increase the viscosity (decrease theReynolds number) and the MVP reaches its maximumincrease at y ≈ , independent of the Reynolds number.The energy transfer of the attached eddies is capturedby the I -integral and we integrate the differential equa-tion given by Eq. (49), from y = 4 . , with the initial y < w > Re= 6 10 Spectral Theory SCT (a) y < w > Re= 10 Spectral TheorySCT (b) -2 y < w > Re= 1.45 10 Spectral TheorySpectral TheorySCT (c) y < w > Re= 2 10 Spectral TheorySCT (d)
FIG. 4. The average of the fluctuation squared as a function of log ( y ) , where y is the distance from the wall (dimensionlessunits). Comparison of experimental data with theory (blue line). condition u = 4 . . This gives the MVP in Fig. 3 (b).This was already done in Ref. [7] and describes how theattached eddies transfer energy into the buffer and theinertial layer. However, we notice that in the predictedMVP over estimates the mean velocity in buffer region.This is because the I -function from Eq. (6) does notaccount for the formation of the attached eddies whichreduce the net energy transfer in the direct cascade.The curves for the fluctuations squared in Fig. 4 areobtained in a similar manner. The attached eddies fixthe peak of (cid:104) w (cid:105) at y ≈ and the peak profiles can befitted by the viscous formula (cid:104) w (cid:105) = a ( y − y ) where a ∼ ( (cid:104) τ o (cid:105) − (cid:104) τ o (cid:105) ) / (cid:104) τ o (cid:105) . This fit is shown in Fig. 4 (c).The peak position is experimentally observed to be fixed,but its height shows a weak Reynolds number dependence a = − .
06 + 0 .
99 log( Re ) , see [21]. This relationship canbe tested using our theory and this will be done in an-other publication, see also [5]. Then, we integrate thedifferential equation from Eq. (50) for w with the ini-tial data described in last section from some point to theright of the peak, where above peak profile fits the initialcondition, this give the profile of the fluctuations squareddown to the flat part in the buffer layer. At the beginningof the flat part, y ≈ , the second scaling from Section II begins to dominate the fluctuations, modeling an in-verse cascade of attached eddies in the buffer layer. Thenwe switch to the buffer I -function I b in the integrationand integrate with I b until we get into the inertial re-gion where the Kolmogorov-Obukhov scaling dominatesagain and the attached eddies break up. This producesthe curves in Fig. 4.We can now compare the functional form of the fluc-tuations squared shown in Fig. 4 with the predictions ofthe stochastic closure theory (SCT) of turbulence, used inRefs. [4] and [12], to compute the Townsend-Perry con-stants, in the inertial (log) layer. These computationsuse the first structure function S of turbulence and weexplain how they are performed, see [4] and [12] for moreinformation. The computed Townsend-Perry constantsare listed in Table I.The first structure function of turbulence is, see [11], E ( | u ( x, t ) − u ( y, t ) | ) = S ( x, y, t )= 2 C (cid:88) k ∈ Z \{ } | d k | (1 − e − λ k t ) | k | ζ + π νC | k | ζ + | sin( πk · ( x − y )) | , where the Reynolds number dependence enters throughthe viscosity ν , and E denotes the expectation (ensamble Re λ C A B A value is computed from C using the proportionality factor A = C / ( K | y ∗ | ζ ) = C / . . average). To get the Kolmogorov-Obukhov coefficients, C p in S p ( r, ∞ ) ∼ C p r ζ p , (51)for the lag variable r small, and ζ p the scaling exponents,we send t to ∞ in the above formulas and project ontothe longitudinal lag variable r = ( r, , . For p = 1 thisbecomes S ∼ π ζ C (cid:88) k (cid:54) =0 | d k | (1 + π νC | k | / ) r ζ = 4 π ζ C ∞ (cid:88) k =1 a ( a + k m )(1 + π νC | k | / ) r ζ , (52)see [11], where ζ = 0 . , see [2]. Now we use the val-ues for ν in Table 1 in [12], and the corresponding valuesfor a, m and C from Table 3 in the same paper. TheReynolds numbers, 6430, 10,770, 15,740 and 19,670 areclose enough to ours 6000, 10,000, 14,500, and 20,000,that we can use value of the parameters in [12]. This givesthe values in Table I, where A ∼ K | y ∗ | ζ C , see SectionV, and the proportionality factor K | y ∗ | ζ = 1 / . iscomputed at the Reynolds number , , where the ap-proximated A coincides with the measured A . The log functions with coefficient A , from the third column inTable I, and using the constant B from the fourth col-umn in Table I, are then compared to the experimentaland theoretical values in Fig. 4. The spanwise Townsend-Perry constants, for the spanwise fluctuations, can com-puted similarly by projecting onto the spanwise lag vari-able t = (0 , t, . In Fig. 4 panel (a), the Townsend-Perry constant A computed by the SCT does not agree with the measuredslope. This was already observed in Ref. [12], since forlow Reynolds numbers the C s do not provide a good ap-proximation to the A s. They only do for large Reynoldsnumbers and the discrepancy (a) occurs at the smallestReynolds number. This does not happen for the General-ized Townsend-Perry constants, the reasons are explainedin Ref. [12], and for them the C p s, p ≥ , provide goodapproximations to the A p s for all Reynolds numbers. VIII. DISCUSSION
We used the spectral theory of the MVP and the vari-ation profile to represent both, and compare with exper-
FIG. 5. Sketch of the instantaneous streaks, in the stream-wise direction, and the wall-attached eddies, in the spanwisedirection. iment [21] for a range of Reynolds numbers. Assumingthat the wall shear stress is a fluctuating quantity, wecan derive that log-law for the variation (2) that was pro-posed by Townsend and measured by Perry and Chong.This law involves the Townsend-Perry constants. Thiswas first done in the large Reynolds number limit andthen for general Reynolds numbers. The Reynolds num-ber dependence of the Townsend-Perry constants is de-termined by the stochastic closure theory [4], [12]. Wederive the log-law for the higher moments of the fluc-tuations and the Generalized Townsend-Perry constantsbased on the functional form of the variation and usethe stochastic closure theory to express them in termsof the Kolmogorov-Obukhov coefficients of the structurefunctions of turbulence [11]. This confirms the results inRefs. [4] and [12].The spectral function I derived in Ref. [7] plays a cen-tral role in this theory. It can be considered be the an-alytic expression of Townsend’s theory of wall-attachededdies. It quantifies when the first eddies appear at theboundary of the viscous and the buffer layer and whenthey are fully developed in the inertial layer. It evenquantifies the limit of their influence in the energeticwake. By introducing the spectral theory into the anal-ysis it resolves many of the issues that we are faced within boundary layer turbulence.The I -function corresponds to the Kolmogorov-Obukhov cascade k − / x in the inertial layer, but in thebuffer layer another cascade k − x dominates the fluctua-tions, although its influence on the MVP is small. This isan inverse cascade that can accelerate larger and largerattached eddies. The energy transfer of this cascade iscaptured by the I -function in buffer layer, I b . With itwe are able to produce the functional form of the aver-aged fluctuations square in the buffer layer. Once in theinertial layer the original I -function dominates again.The final confirmation of this spectral theory is howwe are able to improve the fit to experimental values ofthe MVP in Ref. [7], by use of the I b function in thebuffer layer. Although, this effect on the MVP is small,the attached eddies, siphon a small amount of energyfrom the MVP in the buffer layer. We model this bylinear combination of the I and I b function (1 − a ) I + aI b ,in the buffer layer, where a is small. This produces abetter fit to the measured MVP in the buffer region asshown in Figure 3 (a), whereas the fit without this linearcombination, shown in Figure 3 (b), is not as good.It is fair to ask what the Townsend attached eddiesactually look like since our spectral method is based onthem. Unlike the streamwise streaks and associated vor-tices that have been visualize since the experiments ofKline et al. in the 1960s, see Refs. [13] and [10], theattached eddies are difficult to visualize, either in exper-iments or simulations. We provide a sketch in Fig. 5,where streamwise streaks are visualized gradually lift-ing from the boundary by the flow, and perpendicularto them are spanwise attached eddies being deformed bythe alternating slow and fast streamwise flow into a hair- pin vortex. This does happen both in experiments andobservations, see Ref. [16]. However, these hairpin vor-tices are made unstable by the striations in the stream-wise flow and the typical attached eddies are irregular inshape, with the general feature of being stretched by theflow and attached to the wall. One must interpret theirinfluence in a statistical sense. Acknowledgements:
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